The main result of [1]} roughly states that
\(\operatorname{\mathrm {NP}}^{<e_\chi }(\rho _\chi )\)  and \(\operatorname{\mathrm {HP}}^{<e_\chi }(X_\infty /X)\)  share a vertex
if and only if \(\operatorname{\mathrm {NP}}^{<e_\chi }(\rho ^{\mathrm {ext}}_{P,\chi })\)  and \(\operatorname{\mathrm {HP}}^{<e_\chi }(X^{\mathrm {ext}}_{P,\infty }/\mathbb {A}_{\mathbb {F}_q}^1)\) 
share a corresponding vertex for each \(P \in S\) .
Using the
periodic touching between Newton and Hodge polygons from the previous paragraph,
we obtain periodic touching between \(\operatorname{\mathrm {NP}}^{<e_\chi }(\rho _{\chi })\)  and \(\operatorname{\mathrm {HP}}^{<e_\chi }(X_{\infty }/X)\) .
This immediately allows us to deduce Theorem REF . To establish Theorem REF , we use a classical result on exponential sums (see e.g. [2]}). This result says that \(\operatorname{\mathrm {NP}}^{<e_\chi }(\rho ^{\mathrm {ext}}_{P,\chi })\)  and \(\operatorname{\mathrm {HP}}^{<e_\chi }(X^{\mathrm {ext}}_{P,\infty }/\mathbb {A}_{\mathbb {F}_q}^1)\) 
are equal when \(\chi \)  has order \(p\)  and the congruence conditions of Theorem REF  is satisfied. We then use the work of Kosters-Zhu (and our
equicharacteristic modification) to see that \(\operatorname{\mathrm {NP}}^{<e_\chi }(\rho ^{\mathrm {ext}}_{P,\chi })\)  and \(\operatorname{\mathrm {HP}}^{<e_\chi }(X^{\mathrm {ext}}_{P,\infty }/\mathbb {A}_{\mathbb {F}_q}^1)\)  agree for any finite \(\chi \)  or for \(\chi _0\) .
Theorem REF 
then follows from the main result of [1]}.
